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This is a textbook exercise that might be seen as a generalisation of Pearson's product-moment correlation coefficient between two variables, with multiple groups each having variable sizes:

Let there be $k$ groups of data of size $n_j$ on two variables $(x,y)$, with means $(\bar x_j,\bar y_j)$, variances $(s_{xj}^2,s_{yj}^2)$, and correlation coefficient $r_j$; $j=1,2,\ldots,k$. Then the correlation coefficient of the combined data of size $\sum_{j=1}^kn_j$ is given by

$$r=\frac{\sum_{j=1}^kn_jr_js_{xj}s_{yj}+\sum_{j=1}^k n_j(x_j-\bar x)(y_j-\bar y)}{\sqrt{\sum_{j=1}^kn_js_{xj}^2+\sum_{j=1}^kn_j(\bar x_j-\bar x)}\sqrt{\sum_{j=1}^kn_js_{yj}^2+\sum_{j=1}^kn_j(\bar y_j-\bar y)}}\quad,$$

where $\bar x$ and $\bar y$ are the grand means of $x$ and $y$ respectively.

The exercise also asks to explain from the above formula that $r_j$ may be zero for each $j$ and yet $r$ may be non-zero

While I am not that interested in an analytical derivation of the above expression, I would like to have an intuitive understanding of the fact that the individual correlations may vanish and still the overall correlation may be non-zero. But I am not looking for this verification from the formula. Is there a visual explanation that one can come up with?

By the way, is there a particular name for $r$ as defined above? Any reference where this shows up in descriptive statistics will be great.

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    $\begingroup$ The $r$ defined above is just the ordinary Pearson correlation computed from $x$ and $y$ across all the groups. So it isn't a generalisation and it doesn't have a special name. $\endgroup$ Aug 5, 2018 at 10:26

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This is one option:

There is zero correlation for groups within vertical lines. But overall correlation is positive.

correlation scatter

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  • $\begingroup$ Yea, you are right. Editing to a better example. $\endgroup$
    – yoav_aaa
    Aug 5, 2018 at 12:30

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